This definition is suboptimal, because it forces discontinuityon real line. The definition I recall were continuous(increasing) on real line.

In Abramowitz and Stegun the only definition I have found isvia equations JacobiSn(x, m) = sin(JacobiAmplitude(x, m))and JacobiCn(x, m) = cos(JacobiAmplitude(x, m))

However Mathematica violates those equations:

In[1]:= N[JacobiSN[1+I*2, 3/4], 20]

Out[1]= 1.37533383624444180490 - 0.23845671888060964165 I

In[2]:= N[Sin[JacobiAmplitude[1+I*2, 3/4]], 20]

Out[2]= 1.37533383624444180490 - 0.23845671888060964165 I

In[3]:= N[JacobiCN[1+I*2, 3/4], 20]

Out[3]= -0.33681033299005352886 - 0.97371595177876063808 I

In[4]:= N[Cos[JacobiAmplitude[1+I*2, 3/4]], 20]

Out[4]= -0.33681033299005352886 - 0.97371595177876063808 I

In[5]:= N[JacobiAmplitude[1+I*2, 3/4], 20]

Out[5]= 1.3306295147276587227 - 0.8831325397142208140 I

In[6]:= N[Cos[1.3306295147276587227 - 0.8831325397142208140*I], 20]

Out[6]= 0.336810332990053529 + 0.973715951778760638 I

Note that in Out[4] Mathematica pretends that the equationfor JacobiCN holds, but numerial values in Out[5] and Out[6]show that it is not satisfied. Since the sign of the lastvalue is incorrect and equation for JacobiSN seem to holdone can guess that Mathematica simply usesArcSin[JacobiSN[x, m]] for computing JacobiAmplitude